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Let $A_t$ be the event that $S_1$ is a substring of $S_2$, $S_2=pS_1q$, where the length of $p$ is $t$. Then the probability of $\cup_t A_t$ can be found by inclusion-exclusion as $$\sum P(A_t)-\sum P …

We're looking at the equation $$-\sum_{x\in\mathcal{X}} P(x) \log\left(\frac{Q(x)}{P(x)}\right) = -\sum_{x\in\mathcal{X}} Q(x) \log\left(\frac{P(x)}{Q(x)}\right). $$ In the Bernoulli case, $$-p \log\ …

It seems that in general this is an almost arbitrarily hard problem. Consider the simpler countably infinite case $X=\mathbb N$, $Y=\{0,1\}$. Thus $H=\{x:h(x)=1\}$ is a "random set". Fix $g:X\to\{0, …

It just means that $$\mu\left(\{f: K_f>y\}\right)<\frac{\alpha}{y^\beta}$$ and $$\mu\left(\{f: \rho(x_0,f(x_0))>y\}\right)<\frac{\alpha}{y^\beta}$$

How about if we let $f(t)=\int_0^t W_s^2ds$ where $W$ is a standard Wiener process, and $g(t)=\int_0^tf(s)ds$. Then $g''(t)=W_t^2\ge 0$ so $g$ is convex. For general $C$ replace 0 by $\inf C$.

Since $$E(X_{I_t}\mid I_t)=\mu_{I_t},$$ by iterated expectation $$E(X_{I_t})=E(E(X_{I_t}\mid I_t)) = E(\mu_{I_t}),$$ from which we have the desired equality.

Looks complicated in general, but here's a derivation that for $d=1$ and $k=2$, in the slightly further restricted case where $\phi(X)=aX^2+b$, three moments suffice. Let $m_i=E(\phi(X)^k)$. Recall $ …

Here are some interesting theoretical notions of distance between two such distributions: Wasserstein distance Lévy–Prokhorov metric Total variation distance of probability measures

It's not possible. Let $X$ be constant equal to 1. Let $B_1,B_2,B_3$ be independent Bernoullis. Let $R_1=B_1+B_2$, $C_1=B_3$. Let $R_2=B_1$, $C_2=B_2+B_3$. Then $R_1X+C_1=R_2X+C_2$. So even if yo …

This is not rigorous, but if one constant $a_1$ is small but the others $a_i$ are large, say $a_i\gg n$ for $i>1$ then it seems the $i>1$ terms are negligible and $c\approx X_i^2$, so $\mathrm{Var}(\h …

For any $n$, $$\Pr(Y\le n)=\Pr(X_i\le n\,(\forall i))=\prod_i\Pr(X_i\le n)=\lim_{i\to\infty}\Pr(X_1\le n)^i=0.$$ So $Y=\infty$ with probability 1.

You can use maximum likelihood estimation: https://en.m.wikipedia.org/wiki/Maximum_likelihood_estimation

Does there exist $r$ such that with probability one,$$\limsup_{t \to \infty} {{M_t - m_t}\over{\sqrt{t \log \log t}}} = r?$$ Yes, such an $r$ does exist. First note that by the original LIL, with …

Yes -- Dvoretsky, Erdos and Kakutani 1954 https://books.google.com/books?id=onG8BAAAQBAJ&pg=PA18&lpg=PA18&dq=multiple+points+of+the+Brownian+motion+dense&source=bl&ots=vxQ1n_EC4t&sig=wnDVnqRcN8F1WdCr …

Regarding question 1, the limiting probability of not crossing 0 during the time interval $[1,2^n]$ is $$ \lim_{n \to \infty} \mathbb{P}\{K_n = 0\} = 0 $$ since Brownian motion is recurrent (in dimens …